The symbol $\int$ is known as the **integral sign**. It's used in calculus to denote integration:
$$\int 2x \, dx = x^2 + C$$
$$\int_0^2 x \, dx = 2$$
$$\int_\mathbb{R} \phi(x) \, dx = 1$$
As you can see, it's always followed by a function, and then a differential like $dx$. This indicates which variable is being used in the integration; for instance, $\int f(x,y) \,dx \
eq \int f(x,y) \,dy$, but $\int f(x) \,dx = \int f(t) \,dt$.
Without numbers/variables, it represents the indefinite integral (which is another function); with two limits or with one set as superscripts or subscripts, it denotes the definite integral (which is a number).
You'll see multiple integral signs $\iint$ for an integral with multiple differentials,
$$\iint f(x,y) \,dy \,dx,$$
or a closed integral sign $\oint$ (not sure if that's the correct name) for an integral over a closed curve, surface, etc. Don't expect to see either of those in high school calculus.